Coordinate-energy transformation and the one-point function for the Heisenberg-Ising XXZ spin-1/2 chain on the ring

[2506.14171] Coordinate-energy transformation and the one-point function for the Heisenberg-Ising XXZ spin-1/2 chain on the ring Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematical Physics arXiv:2506.14171 (math-ph) [Submitted on 17 Jun 2025 ( v1 ), last revised 14 Aug 2026 (this version, v3)] Title: Coordinate-energy transformation and the one-point function for the Heisenberg-Ising XXZ spin-1/2 chain on the ring Authors: Eric I. Corwin , Nikolaus Elsaesser , Axel Saenz View a PDF of the paper titled Coordinate-energy transformation and the one-point function for the Heisenberg-Ising XXZ spin-1/2 chain on the ring, by Eric I. Corwin and 2 other authors View PDF HTML (experimental) Abstract: We provide explicit formulas to diagonalize the Hamiltonian for the Heisenberg-Ising XXZ spin-1/2 chain on a discrete ring. Two distinguished bases for the Hilbert space include the basis labeled by the coordinates of the particle configurations and the basis obtained from the eigenvectors of the Hamiltonian. We diagonalize the Hamiltonian by providing an explicit transformation between these two distinguished bases. The transformation from the coordinate basis to the eigenbasis is given by the well-known coordinate Bethe Ansatz. Our contribution is the transformation from the eigenbasis to the coordinate basis, which we call the inverse coordinate Bethe Ansatz transformation/formula. We prove that the inverse coordinate Bethe Ansatz transformation is indeed the inverse of the transformation obtained from the Bethe Ansatz for the case of N = 2 particles and a ring of odd length L with a small nonzero anisotropy term $\Delta < (L-1)/(2L)$ and $\Delta$ outside some exceptional finite set. The case of N > 2 particles and a ring of odd length L is numerically confirmed for different arbitrary choices of parameters and is left as a conjecture. Additionally, assuming that the conjecture is true, we derive an exact formula for the one-point function of the system through special identities for the Izergin-Korepin determinant. Moreover, if the conjecture is true, this implies that the Bethe Ansatz is complete. Comments: 49 pages, 1 figure V2 - N=2 proof included V3 - typos corrected Subjects: Mathematical Physics (math-ph) ; Probability (math.PR) MSC classes: 82B23 (Primary) 82B20, 60K35 (Secondary) Cite as: arXiv:2506.14171 [math-ph] (or arXiv:2506.14171v3 [math-ph] for this version) https://doi.org/10.48550/arXiv.2506.14171 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Axel Saenz [ view email ] [v1] Tue, 17 Jun 2025 04:26:52 UTC (1,055 KB) [v2] Fri, 7 Aug 2026 11:43:50 UTC (1,078 KB) [v3] Fri, 14 Aug 2026 22:32:46 UTC (1,078 KB) Full-text links: Access Paper: View a PDF of the paper titled Coordinate-energy transformation and the one-point function for the Heisenberg-Ising XXZ spin-1/2 chain on the ring, by Eric I. 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